296edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 56: | Line 48: | ||
| 0.2206 | | 0.2206 | ||
| 5.44 | | 5.44 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 92: | Line 78: | ||
|- | |- | ||
| 8 | | 8 | ||
| 144\296<br>(4\296) | | 144\296<br />(4\296) | ||
| 583.78<br>(16.22) | | 583.78<br />(16.22) | ||
| 7/5<br>(126/125) | | 7/5<br />(126/125) | ||
| [[Octoid]] | | [[Octoid]] | ||
|- | |- | ||
| 37 | | 37 | ||
| 67\296<br>(3\296) | | 67\296<br />(3\296) | ||
| 271.62<br>(12.16) | | 271.62<br />(12.16) | ||
| 117/100<br>(?) | | 117/100<br />(?) | ||
| [[Dzelic]] | | [[Dzelic]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
[[Category:Sabric]] | [[Category:Sabric]] | ||
Revision as of 04:24, 16 November 2024
| ← 295edo | 296edo | 297edo → |
Theory
In the 5-limit, 296et not only tempers out the semicomma of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its optimal patent val, and tempers out the minortone comma, [-16 35 -17⟩. It is also an interesting temperament in higher limits, being distinctly consistent through to the 15-odd-limit. In the 7-limit it tempers out 4375/4374 (ragisma), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), supporting 7-limit octoid and sabric. In the 11-limit, 540/539, 1375/1372, 3025/3024, 4000/3993, 6250/6237 and 9801/9800; in the 13-limit, 625/624, 729/728, 1575/1573, 1716/1715, 2080/2079, and 6656/6655, so that it also supports the 11- and 13-limit versions of octoid. It allows swetismic chords and squbemic chords in the 13-odd-limit, in addition to nicolic chords in the 15-odd-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.60 | -1.18 | +0.09 | +0.03 | -1.34 | +0.45 | -1.57 | +0.10 | +0.15 | -1.79 |
| Relative (%) | +0.0 | -14.9 | -29.1 | +2.3 | +0.8 | -33.0 | +11.1 | -38.7 | +2.6 | +3.8 | -44.2 | |
| Steps (reduced) |
296 (0) |
469 (173) |
687 (95) |
831 (239) |
1024 (136) |
1095 (207) |
1210 (26) |
1257 (73) |
1339 (155) |
1438 (254) |
1466 (282) | |
Subsets and supersets
Since 296 factors into 23 × 37, 296edo has subset edos 2, 4, 8, 37, 74 and 148.
Regular temperament properties
Template:Comma basis begin |- | 2.3 | [-469 296⟩ | [⟨296 469]] | +0.1904 | 0.1905 | 4.70 |- | 2.3.5 | 2109375/2097152, [-16 35 -17⟩ | [⟨296 469 687]] | +0.2962 | 0.2158 | 5.32 |- | 2.3.5.7 | 4375/4374, 16875/16807, 2100875/2097152 | [⟨296 469 687 831]] | +0.2138 | 0.2350 | 5.80 |- | 2.3.5.7.11 | 540/539, 1375/1372, 4000/3993, 2100875/2097152 | [⟨296 469 687 831 1024]] | +0.1691 | 0.2284 | 5.63 |- | 2.3.5.7.11.13 | 540/539, 625/624, 729/728, 1375/1372, 15379/15360 | [⟨296 469 687 831 1024 1095]] | +0.2012 | 0.2206 | 5.44 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 45\296
| 182.43
| 10/9
| Mitonic
|-
| 1
| 67\296
| 271.62
| 75/64
| Sabric
|-
| 1
| 105\296
| 425.68
| 2625/2048
| Rainwell
|-
| 2
| 57\296
| 231/08
| 8/7
| Orga
|-
| 8
| 144\296
(4\296)
| 583.78
(16.22)
| 7/5
(126/125)
| Octoid
|-
| 37
| 67\296
(3\296)
| 271.62
(12.16)
| 117/100
(?)
| Dzelic
Template:Rank-2 end
Template:Orf